The Hill function is the universal Hopfield barrier for sharpness of input–output responses

Significance

Biological systems can process information without expending energy, and the limit to what can be achieved in this way is known as a Hopfield barrier. We characterize this barrier for the sharpness of input–output responses, making typical assumptions about the underlying molecular mechanisms. If an input ligand binds at $N$ sites, we show that the Hopfield barrier for sharpness is the Hill function with coefficient $N$ , irrespective of the molecular details. This provides a biophysical justification for the widely used Hill function, which was introduced over a century ago only as an empirical fit to data. Furthermore, when data exceed the sharpness barrier, the strong conclusion may be drawn that the underlying mechanism is expending energy.

Abstract

The Hill functions, ${\mathcal{H}}_h(x)=x^h/(1+x^h)$ , have been widely used in biology for over a century but, with the exception of ${\mathcal{H}}_1$ , they have had no justification other than as a convenient fit to empirical data. Here, we show that they are the universal limit for the sharpness of any input–output response arising from a Markov process model at thermodynamic equilibrium. Models may represent arbitrary molecular complexity, with multiple ligands, internal states, conformations, coregulators, etc, under core assumptions that are detailed in the paper. The model output may be any linear combination of steady-state probabilities, with components other than the chosen input ligand held constant. This formulation generalizes most of the responses in the literature. We use a coarse-graining method in the graph-theoretic linear framework to show that two sharpness measures for input–output responses fall within an effectively bounded region of the positive quadrant, ${\mathrm{\Omega }}_m\subset {({\mathbb{R}}^{+})}^2$ , for any equilibrium model with $m$ input binding sites. ${\mathrm{\Omega }}_m$ exhibits a cusp which approaches, but never exceeds, the sharpness of ${\mathcal{H}}_m$ , but the region and the cusp can be exceeded when models are taken away from thermodynamic equilibrium. Such fundamental thermodynamic limits are called Hopfield barriers, and our results provide a biophysical justification for the Hill functions as the universal Hopfield barriers for sharpness. Our results also introduce an object, ${\mathrm{\Omega }}_m$ , whose structure may be of mathematical interest, and suggest the importance of characterizing Hopfield barriers for other forms of cellular information processing.